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| In [[astronomy]], '''perturbation''' is the complex motion of a massive body subject to forces other than the gravitational attraction of a single other massive body.<ref name="BMW"> | | In [[geometry]], an '''intrinsic equation''' of a curve is an equation that defines the curve using a relation between the curve's intrinsic properties, that is, properties that do not depend on the location and possibly the orientation of the curve. Therefore an intrinsic equation defines the shape of the curve without specifying its position relative to an arbitrarily defined coordinate system. |
| {{cite book
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| |last1 = Bate
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| |first1 = Roger R.
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| |last2 = Mueller
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| |first2 = Donald D.
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| |last3 = White
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| |first3 = Jerry E.
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| | title = Fundamentals of Astrodynamics
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| |publisher = Dover Publications, Inc., New York
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| |isbn = 0-486-60061-0
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| |date=1971}}, e.g. at ch. 9, p. 385.</ref>
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| The other forces can include a third (fourth, fifth, etc.) body, [[Drag_(physics)|resistance]], as from an [[atmosphere]], and the off-center attraction of an [[Oblate_spheroid|oblate]] or otherwise misshapen body.<ref name="moulton"/>
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| [[File:Moon perturbation diagram.PNG|thumb|300px|alt=Vector diagram of the Sun's perturbations on the Moon. When the gravitational force of the Sun common to both the Earth and the Moon is subtracted, what is left is the perturbations.|The perturbing forces of the [[Sun]] on the [[Moon]] at two places in its [[orbit]]. The blue arrows represent the [[Euclidean_vector|direction and magnitude]] of the gravitational force on the [[Earth]]. Applying this to both the Earth's and the Moon's position does not disturb the positions relative to each other. When it is subtracted from the force on the Moon (black arrows), what is left is the perturbing force (red arrows) on the Moon relative to the Earth. Because the perturbing force is different in direction and magnitude on opposite sides of the orbit, it produces a change in the shape of the orbit.]] | | The intrinsic quantities used most often are [[arc length]] <math> s </math>, [[tangential angle]] <math> \theta </math>, [[curvature]] <math>\kappa</math> or [[Radius of curvature (mathematics)|radius of curvature]], and, for 3-dimensional curves, [[Torsion of a curve|torsion]] <math>\tau </math>. Specifically: |
| | * The [[natural equation]] is the curve given by its curvature and torsion. |
| | * The [[Whewell equation]] is obtained as a relation between arc length and tangential angle. |
| | * The [[Cesàro equation]] is obtained as a relation between arc length and curvature. |
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| == Introduction ==
| | The equation of a circle (including a line) for example is given by the equation <math>\kappa(s) = \tfrac{1}{r}</math> where <math>s</math> is the arc length, <math>\kappa</math> the curvature and <math>r</math> the radius of the circle. |
| The study of perturbations began with the first attempts to predict planetary motions in the sky, although in ancient times the causes remained a mystery. [[Isaac Newton|Newton]], at the time he formulated his laws of [[Newton's laws of motion|motion]] and of [[Newton's law of universal gravitation|gravitation]], applied them to the first analysis of perturbations,<ref name="moulton"/> recognizing the complex difficulties of their calculation.<ref>Newton in 1684 wrote: "By reason of the deviation of the Sun from the center of gravity, the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. Each time a planet revolves it traces a fresh orbit, as in the motion of the Moon, and each orbit depends on the combined motions of all the planets, not to mention the action of all these on each other. But to consider simultaneously all these causes of motion and to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind." (quoted by Prof G E Smith (Tufts University), in [http://google.com/search?q=cache:8RItNNOcJJoJ:www.stanford.edu/dept/cisst/SmithPowerpointTalk1.ppt "Three Lectures on the Role of Theory in Science"] 1. Closing the loop: Testing Newtonian Gravity, Then and Now); and Prof R F Egerton (Portland State University, Oregon) after quoting the same passage from Newton concluded: [http://physics.pdx.edu/~egertonr/ph311-12/newton.htm "Here, Newton identifies the "many body problem" which remains unsolved analytically."]</ref> Many of the great mathematicians since then have given attention to the various problems involved; throughout the 18th and 19th centuries there was demand for accurate tables of the position of the Moon and planets for purposes of navigation at sea. | |
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| The complex motions of gravitational perturbations can be broken down. The hypothetical motion that the body follows under the gravitational effect of one other body only is typically a [[conic section]], and can be readily described with the methods of [[geometry]]. This is called a [[two-body problem]], or an unperturbed [[Kepler orbit|Keplerian orbit]]. The differences between that and the actual motion of the body are '''perturbations''' due to the additional gravitational effects of the remaining body or bodies. If there is only one other significant body then the perturbed motion is a [[three-body problem]]; if there are multiple other bodies it is an [[n-body problem|''n''-body problem]]. Analytical solutions (mathematical expressions to predict the positions and motions at any future time) for the two-body and three-body problems exist; none has been found for the ''n''-body problem except for certain special cases. Even the two-body problem becomes insoluble if one of the bodies is irregular in shape.<ref name="roy">{{cite book |last = Roy |first = A.E. | title = Orbital Motion |publisher = Institute of Physics Publishing |edition = third |isbn = 0-85274-229-0 |date=1988}}, chapters 6 and 7.</ref>
| | These coordinates greatly simplify some physical problem. For elastic rods for example, the potential energy is given by |
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| | :<math>E= \int_0^L B \kappa^2(s)ds </math> |
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| [[File:Mercury perturbation comparison.png|thumb|300px|alt=Plot of Mercury's position in its orbit, with and without perturbations from various planets. The perturbations cause Mercury to move in looping paths around its unperturbed position.|[[Mercury (planet)|Mercury's]] orbital longitude and latitude, as perturbed by [[Venus]], [[Jupiter]] and all of the planets of the [[Solar System]], at intervals of 2.5 days. Mercury would remain centered on the crosshairs if there were no perturbations.]]
| | where <math>B</math> is the bending modulus <math>EI</math>. Moreover, as <math>\kappa(s) = d\theta/ds</math>, elasticity of rods can be given a simple [[Calculus of variations|variational]] form. |
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| Most systems that involve multiple gravitational attractions present one primary body which is dominant in its effects (for example, a star, in the case of the star and its planet, or a planet, in the case of the planet and its satellite). The gravitational effects of the other bodies can be treated as perturbations of the hypothetical unperturbed motion of the planet or satellite around its primary body.
| | ==References== |
| | * {{cite book | author=R.C. Yates | title=A Handbook on Curves and Their Properties | location=Ann Arbor, MI | publisher=J. W. Edwards | pages=123–126 | year=1952 }} |
| | * {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=1–5 }} |
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| == Mathematical analysis == | | ==External links== |
| | *{{MathWorld |title=Intrinsic Equation |urlname=IntrinsicEquation}} |
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| === General perturbations ===
| | [[Category:Curves]] |
| In methods of '''general perturbations''', general differential equations, either of motion or of change in the [[orbital elements]], are solved analytically, usually by [[series expansion]]s. The result is usually expressed in terms of algebraic and trigonometric functions of the orbital elements of the body in question and the perturbing bodies. This can be applied generally to many different sets of conditions, and is not specific to any particular set of gravitating objects.<ref>Bate, Mueller, White (1971), e.g. at p.387 and at section 9.4.3, p.410.</ref> Historically, general perturbations were investigated first. The classical methods are known as ''variation of the elements'', ''[[variation of parameters]]'' or ''variation of the constants of integration''. In these methods, it is considered that the body is always moving in a [[conic section]], however the conic section is constantly changing due to the perturbations. If all perturbations were to cease at any particular instant, the body would continue in this (now unchanging) conic section indefinitely; this conic is known as the [[osculating orbit]] and its [[orbital elements]] at any particular time are what are sought by the methods of general perturbations.<ref name="moulton"/>
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| General perturbations takes advantage of the fact that in many problems of [[celestial mechanics]], the two-body orbit changes rather slowly due to the perturbations; the two-body orbit is a good first approximation. General perturbations is applicable only if the perturbing forces are about one order of magnitude smaller, or less, than the gravitational force of the primary body.<ref name="roy"/> In the [[Solar System]], this is usually the case; [[Jupiter]], the second largest body, has a mass of about 1/1000 that of the [[Sun]].
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| General perturbation methods are preferred for some types of problems, as the source of certain observed motions are readily found. This is not necessarily so for special perturbations; the motions would be predicted with similar accuracy, but no information on the configurations of the perturbing bodies (for instance, an [[orbital resonance]]) which caused them would be available.<ref name="roy"/>
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| === Special perturbations ===
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| In methods of '''special perturbations''', numerical datasets, representing values for the positions, velocities and accelerative forces on the bodies of interest, are made the basis of [[numerical integration]] of the differential [[equations of motion]].<ref>Bate, Mueller, White (1971), pp.387-409.</ref> In effect, the positions and velocities are perturbed directly, and no attempt is made to calculate the curves of the orbits or the [[orbital elements]].<ref name=moulton>
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| {{cite web
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| |url=http://books.google.com/books?id=jqM5AAAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
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| |title=An Introduction to Celestial Mechanics, Second Revised Edition
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| |last=Moulton
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| |first=Forest Ray
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| |year=1914}} chapter IX. (at [http://books.google.com/books Google books])</ref>
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| Special perturbations can be applied to any problem in [[celestial mechanics]], as it is not limited to cases where the perturbing forces are small.<ref name="roy"/> Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generated [[Fundamental_ephemeris|planetary ephemerides]] of the great astronomical almanacs.<ref name="moulton"/><ref>
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| see, for instance, [[Jet Propulsion Laboratory Development Ephemeris]]</ref>
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| ==== Cowell's method ====
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| [[File:Cowells method.png|thumb|Cowell's method. Forces from all perturbing bodies (black and gray) are summed to form the total force on body ''i'' (red), and this is numerically integrated starting from the initial position (the ''epoch of osculation'').]]
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| is perhaps the simplest of the special perturbation methods;<ref>
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| So named for [[Philip_Herbert_Cowell|Philip H. Cowell]], who, with A.C.D. Cromellin, used a similar method to predict the return of Halley's comet.
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| {{cite book
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| |last1 = Brouwer
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| |first1 = Dirk
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| |last2 = Clemence
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| |first2 = Gerald M.
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| | title = Methods of Celestial Mechanics
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| |publisher = Academic Press, New York and London
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| |date=1961}}, p. 186.
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| </ref>
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| mathematically, for <math>n</math> mutually interacting bodies, [[Newton's_law_of_universal_gravitation|Newtonian]] forces on body <math>i</math> from the other bodies <math>j</math> are simply summed thus,
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| : <math>\mathbf{\ddot{r}}_i = \sum_{\underset{j \ne i}{j=1}}^n {Gm_j (\mathbf{r}_j-\mathbf{r}_i) \over r_{ij}^3}</math>
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| where <math>\mathbf{\ddot{r}}_i</math> is the [[acceleration]] vector of body <math>i</math>, <math>G</math> is the [[gravitational constant]], <math>m_j</math> is the [[mass]] of body <math>j</math>, <math>\mathbf{r}_i</math> and <math>\mathbf{r}_j</math> are the [[Position_vector|position vectors]] of objects <math>i</math> and <math>j</math> and <math>r_{ij}</math> is the distance from object <math>i</math> to object <math>j</math>, all [[Euclidean_vector#Physics|vectors]] being referred to the [[Center_of_mass#Astronomy|barycenter]] of the system. This equation is resolved into components in <math>x</math>, <math>y</math>, <math>z</math> and these are integrated numerically to form the new velocity and position vectors. This process is repeated as many times as necessary. The advantage of Cowell's method is ease of application and programming. A disadvantage is that when perturbations become large in magnitude (as when an object makes a close approach to another) the errors of the method also become large.<ref name="danby">
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| {{cite book
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| |last = Danby
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| |first = J.M.A.
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| | title = Fundamentals of Celestial Mechanics
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| |publisher = Willmann-Bell, Inc.
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| |edition = second |isbn = 0-943396-20-4
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| |date=1988}}, chapter 11.</ref>
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| For many problems in [[celestial mechanics]], this is never the case. Another disadvantage is that in systems with a dominant central body, such as the [[Sun]], it is necessary to carry many [[Significant figures|significant digits]] in the [[arithmetic]] because of the large difference in the forces of the central body and the perturbing bodies, although with modern [[computer]]s this is not nearly the limitation it once was.<ref>
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| {{cite book
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| |last = Herget
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| |first = Paul
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| | title = The Computation of Orbits
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| |publisher = privately published by the author
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| |date=1948}}, p. 91 ff.</ref>
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| ==== Encke's method ====
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| [[File:Enckes method.PNG|thumb|Encke's method. Greatly exaggerated here, the small difference δ'''r''' (blue) between the osculating, unperturbed orbit (black) and the perturbed orbit (red), is numerically integrated starting from the initial position (the ''epoch of osculation'').]]
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| begins with the [[osculating orbit]] as a reference and integrates numerically to solve for the variation from the reference as a function of time.<ref>
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| So named for [[Johann Franz Encke]];
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| {{cite book
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| |last = Battin
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| |first = Richard H.
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| | title = An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition
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| |publisher = American Institute of Aeronautics and Astronautics, Inc.
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| |isbn = 1-56347-342-9
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| |date=1999}}, p. 448</ref>
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| Its advantages are that perturbations are generally small in magnitude, so the integration can proceed in larger steps (with resulting lesser errors), and the method is much less affected by extreme perturbations. Its disadvantage is complexity; it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as ''rectification''.<ref name="danby"/> Encke's method is similar to the general perturbation method of variation of the elements, except the rectification is performed at discrete intervals rather than continuously.<ref>
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| Battin (1999), sec. 10.2.</ref>
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| Letting <math>\boldsymbol{\rho}</math> be the [[Position_vector|radius vector]] of the [[osculating orbit]], <math>\mathbf{r}</math> the radius vector of the perturbed orbit, and <math>\delta \mathbf{r}</math> the variation from the osculating orbit,
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| {{NumBlk|:|<math>\delta \mathbf{r} = \mathbf{r} - \boldsymbol{\rho}</math>, and the [[Equations_of_motion|equation of motion]] of <math>\delta \mathbf{r}</math> is simply|{{EquationRef|1}}}}
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| {{NumBlk|:|<math>\ddot{\delta \mathbf{r}} = \mathbf{\ddot{r}} - \boldsymbol{\ddot{\rho}}</math>.|{{EquationRef|2}}}}
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| <math>\mathbf{\ddot{r}}</math> and <math>\boldsymbol{\ddot{\rho}}</math> are just the equations of motion of <math>\mathbf{r}</math> and <math>\boldsymbol{\rho}</math>,
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| {{NumBlk|:|<math>\mathbf{\ddot{r}} = \mathbf{a}_{\text{per}} - {\mu \over r^3} \mathbf{r}</math> for the perturbed orbit and |{{EquationRef|3}}}}
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| {{NumBlk|:|<math>\boldsymbol{\ddot{\rho}} = - {\mu \over \rho^3} \boldsymbol{\rho}</math> for the unperturbed orbit,|{{EquationRef|4}}}}
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| where <math>\mu = G(M+m)</math> is the [[Standard gravitational parameter|gravitational parameter]] with <math>M</math> and <math>m</math> the [[mass]]es of the central body and the perturbed body, <math>\mathbf{a}_{\text{per}}</math> is the perturbing [[acceleration]], and <math>r</math> and <math>\rho</math> are the magnitudes of <math>\mathbf{r}</math> and <math>\boldsymbol{\rho}</math>.
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| Substituting from equations ({{EquationNote|3}}) and ({{EquationNote|4}}) into equation ({{EquationNote|2}}),
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| {{NumBlk|:|<math>\ddot{\delta \mathbf{r}} = \mathbf{a}_{\text{per}} + \mu \left( {\boldsymbol{\rho} \over \rho^3} - {\mathbf{r} \over r^3} \right)</math>, |{{EquationRef|5}}}}
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| which, in theory, could be integrated twice to find <math>\delta \mathbf{r}</math>. Since the osculating orbit is easily calculated by two-body methods, <math>\boldsymbol{\rho}</math> and <math>\delta \mathbf{r}</math> are accounted for and <math>\mathbf{r}</math> can be solved. In practice, the quantity in the brackets, <math> {\boldsymbol{\rho} \over \rho^3} - {\mathbf{r} \over r^3} </math>, is the difference of two nearly equal vectors, and further manipulation is necessary to avoid the need for extra [[Significant figures|significant digits]].<ref>
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| Bate, Mueller, White (1971), sec. 9.3.</ref><ref>
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| Roy (1988), sec. 7.4.</ref>
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| Encke's method was more widely used before the advent of modern [[computer]]s, when much orbit computation was performed on [[Calculating_machine|mechanical calculating machines]].
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| == Periodic nature ==
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| [[File:Eccentricity rocky planets.jpg|thumb|300px|[http://www.orbitsimulator.com/gravity/articles/what.html Gravity Simulator] plot of the changing [[orbital eccentricity]] of [[Mercury (planet)|Mercury]], [[Venus]], [[Earth]], and [[Mars]] over the next 50,000 years. The 0 point on this plot is the year 2007.]]
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| In the Solar System, many of the disturbances of one planet by another are periodic, consisting of small impulses each time a planet passes another in its orbit. This causes the bodies to follow motions that are periodic or quasi-periodic – such as the Moon in its [[Lunar theory|strongly perturbed]] [[Orbit of the Moon|orbit]], which is the subject of [[lunar theory]]. This periodic nature led to the [[discovery of Neptune]] in 1846 as a result of its perturbations of the orbit of [[Uranus]].
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| On-going mutual perturbations of the planets cause long-term quasi-periodic variations in their [[orbital element]]s, most apparent when two planets' orbital periods are nearly in sync. For instance, five orbits of [[Jupiter]] (59.31 years) is nearly equal to two of [[Saturn]] (58.91 years). This causes large perturbations of both, with a period of 918 years, the time required for the small difference in their positions at [[Conjunction_(astronomy_and_astrology)|conjunction]] to make one complete circle, first discovered by [[Pierre-Simon_Laplace|Laplace]].<ref name="moulton"/> [[Venus]] currently has the orbit with the least [[Orbital eccentricity|eccentricity]], i.e. it is the closest to [[Circle|circular]], of all the planetary orbits. In 25,000 years' time, [[Earth]] will have a more circular (less eccentric) orbit than Venus. It has been shown that long-term periodic disturbances within the [[Solar System]] can become chaotic over very long time scales; under some circumstances one or more [[planet]]s can cross the orbit of another, leading to collisions.<ref>see references at [[Stability of the Solar System]]</ref>
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| The orbits of many of the minor bodies of the Solar System, such as [[comet]]s, are often heavily perturbed, particularly by the gravitational fields of the [[gas giant]]s. While many of these perturbations are periodic, others are not, and these in particular may represent aspects of [[chaotic motion]]. For example, in April 1996, [[Jupiter]]'s gravitational influence caused the [[Orbital period|period]] of [[Comet Hale–Bopp]]'s orbit to decrease from 4,206 to 2,380 years, a change that will not revert on any periodic basis.<ref name=perturb>{{cite web
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| |date=1997-04-10
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| |title=Comet Hale-Bopp Orbit and Ephemeris Information
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| |publisher=JPL/NASA
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| |author=Don Yeomans
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| |url=http://www2.jpl.nasa.gov/comet/ephemjpl8.html
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| |accessdate=2008-10-23}}</ref>
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| == See also ==
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| * [[Nereid (moon)|Nereid]] one of the outer moons of Neptune with a high [[orbital eccentricity]] of ~0.75 and is frequently perturbed
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| * [[Osculating orbit]]
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| * [[Orbital resonance]]
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| * [[Stability of the Solar System]]
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| * [[Formation and evolution of the Solar System]]
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| == External links ==
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| * [http://main.chemistry.unina.it/~alvitagl/solex/MarsDist.html Solex] (by Aldo Vitagliano) predictions for the position/orbit/close approaches of Mars
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| * [http://books.google.com/books?id=snK4AAAAIAAJ&source=gbs_navlinks_s Gravitation] Sir George Biddell Airy's 1884 book on gravitational motion and perturbations, using little or no math. A good source if you can stand the flowery 19th-century English. (at [http://books.google.com/books Google books])
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| == Notes and references ==
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| <references/>
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| [[Category:Orbital perturbations]] | |
| [[Category:Dynamical_systems]]
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| [[Category:Dynamics_of_the_Solar_System]]
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